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Toward Spectral Sequences 3: Cell Complexes and Ordinary Homology
Topic CW complexes and ordinary (cellular) homology. Part of the series outlined at Towards Spectral Sequences. Seminar so last time we talked about the homotopy groups of a space and fibrations (things that lifted homotopies) and how they fit together this time we're going to cover CW complexes, which i promised last time and then didn't deliver on, go through some facts about their homotopic behavior, and start looking at homology algebraic topology started life as combinatorial topology, where people studied how you could build spaces out of smaller pieces and what the resultant topology looked like and a CW complex is a particular example of such a thing so to start, the goal here is to come up with a general way to build up spaces in an inductive manner such that maps out of the various inductive steps correspond to maps out of the finished object and so what you ought to expect to see are a bunch of colimits ah, hell, let's just jump in. a CW complex begins life as a discrete space, i.e. a collection of points these things we'll call 0-cells, and the space the 0-skeleton of the complex given the n-skeleton, we construct the (n+1)-skeleton by pushing the n-skeleton out along a bunch of (n+1)-cells the relevant diagram looks like: hmm, i'm missing a coproduct symbol on the upper right. the point is that we attach a bunch of (n+1)-disks to our n-skeleton by gluing along maps that assign the boundaries of the new n+1-disks to points in the n-skeleton if this terminates after finitely many dimensions (i.e. we attach no cells of dimension greater than some n), we call that object the resultant CW-complex if it doesn't, we take a colimit to make topology place nice, we also require that the space be locally finite (any point in the CW-complex is contained in the image of finitely many attaching maps of any particular dimension) sometimes you'll hear that CW-complexes have the "weak topology", which states that a set is closed in the complex iff its intersection is closed in the subspace topology for each cell we don't have to worry about this because the colimit takes care of that proving that maps out of the complex correspond to maps out of the cells is very easy, and i'll leave it (as ever) as an exercise while we're here, we should take some time to talk about cofibrations and relative homotopy groups a fibration, as you'll recall, was an epimorphism p: E -> B such that a homotopy X * I -> B and an initial lift X -> E extended to a lifted homotopy X * I -> E sometimes just called the homotopy lifting property if you dualize the relevant diagram, you get this: the property communicated here is that we have a subspace A of a space X and a homotopy of A in Y, along with an initial extension of this homotopy as a homotopy of X in Y and from that information we can complete the extension and come up with a homotopy of X in Y that agrees with the old homotopy of A in Y sometimes called the homotopy extension property i don't offhand have any great examples of things that aren't cofibrations, but i do have a good example of things that are cofibrations: any subcomplex A of a CW-complex X is a cofibration in the light of our earlier investigation of relative homotopy groups, if A -> X is a cofibration and A is contractible, then pi_n(X, A) = pi_n(X / A) just to give you some idea of what we'd use these objects for we also ought to construct some CW-complexes as examples, i suppose an n-sphere can be turned into a CW complex in two different ways, both of which have their uses first, we can take the 0 through (n-1)-skeletons to be just a single point, then attach an n-cell to this point, and then terminate that's perhaps the obvious one the still-obvious-but-slightly-less-so complex is to start with a pair of points, representative of S^0 to form S^1, we attach two 1-cells, both with one endpoint on one point and the other endpoint on the other endpoint to form S^2, we attach two 2-cells, again both of whose attaching maps should be the identity S^1 -> S^1 and so on. we construct S^n for S^(n-1) by attaching two n-cells in the same manner we can construct a torus by taking a single point, attaching two 1-cells so it looks like a wedge sum of two circles (these should be thought of as one lying along a meridian and the other along a latitudinal ring) then attaching one 2-cell by walking all the way around the rectangle formed (so around the latitudinal ring, up the meridian, around the latitudinal ring the other way, down the meridian) there are, of course, other CW-structures we can put on the torus, but this one is nice and tiny as you'll recall, the torus is actually the product of two S^1s in general, under nice conditions (both complexes are finite, iirc), the product of two complexes will again be a complex the "nice conditions" part is just to make the colimit behave well we can also give a complex structure for RP^2, which we discussed a little bit last time we take a single point, attach a 1-cell to it to form a hoop, and then attach a 2-cell by walking around the hoop twice this is a little difficult to visualize, but it gives a nice idea of what pi_1 RP^2 looks like if we take a loop in RP^2 that lies along the 1-cell, it's not contractible (try thinking of what it would look like to lift it up onto the 2-cell; you can't lift the beginning and end of the loop to the same place) but if we follow the 1-cell around /twice/, we do get something contractible, by construction and so pi_1 RP^2 is Z_2 (surjectivity of this is not difficult to see, and something we'll come back to in a more general setting) this is probably enough. there's another kind of complex, a simplicial complex to start talking about those, we'll need to know what simplices (singular: simplex) are i define the 0-simplex Delta^0 to be a one-point space, then inductively define Delta^n to be (Delta^(n-1) * I) / ((x, 0) ~ (y, 0)) for all x, y in Delta^(n-1) (so an unreduced cone) like the sequence of spheres forms a lot of nice, familiar spaces, so do these simplices we have a point, then a line, then a filled triangle, then a filled tetrahedron, and so on these also have nice combinatorial properties; the n-simplex contains (n+1) (n-1)-simplices as "faces", subspaces sitting on the exterior of the figure just like with CW-complexes, we can build simplicial complexes by taking the 0-skeleton to be a collection of points, then attaching n-simplices to the (n-1)-skeleton with pushouts whose attaching maps stick the boundaries of each n-simplex into existing (n-1)-simplices in the (n-1)-skeleton the reason i bring these structures up is to talk about simplicial approximation, which is a lemma that reads as follows: For a CW-pair (X, A) where X is obtained from A by attaching a single n-cell e^n and a function f: (K, L) -> (X, A) where (K, L) is a relative pair of finite simplicial complexes, there is a subdivision (K', L') of (K, L) and a map f' homotopy to f relative to f^{-1}(A) (on which they agree), and if sigma is a simplex in K' with f'(sigma) cap e^n nonempty, f'(sigma) is contained in the interior of e^n and f' is locally linear on sigma the idea is that attaching one cell to a CW-complex can be done /very/ nicely. if we already have a simplicial structure on the rest of a CW complex, then we can come up with something so close to attaching an n-cell that we don't care about it that corresponds to attaching an n-simplex and is linear ! this gives us the tool we need to prove a wide variety of results about the homotopy of CW-complexes yeah, go ahead what's "linear"? we identify D^n with the unit ball in R^n, where linearity makes sense sufficient answer? yeah, go on, I'll think about it okay anyway, lots of results about homotopy of CW-complexes the first is a pretty direct application of the lemma: we say a pair (X, A) is a relative CW-pair when A is a CW-complex and X is formed from A by attaching cells ! yes, go ahead is a relative CW-pair different from a relative pair in the lemma's statement? no, it's just looking at it from a different direction ok we'll say that the n-skeleton of (X, A) is A union X's n-skeleton the pair (X, n-skel of (X, A)) then has vanishing relative ith homotopy groups for i <= n this in turn gives us that the inclusion of X's n-skeleton into X induces isomorphisms in homotopy groups less than or equal to n an epimorphism at n (use the l.e.s. of relative homotopy) so, finally, we can say with confidence that pi_m S^n = 0 for m < n using the first CW structure for S^n that i gave for those familiar with homology (and those of you who aren't will become familiar quite shortly), there's an axiom called the excision axiom that gives sufficient conditions on a triple of spaces (X, A, B) with A cup B = X to induce an isomorphism H_*(A, A cap B) -> H_*(X, B) we can abuse the simplicial approximation lemma to get a similar, though weaker, result in homotopy to this end, we define the relative pathspace P(X, A) of a pointed space X to be the subspace of X^I with f in X^I satisfying f(0) = x0 (i.e. the map is pointed) and f(1) lies in A we'll take X, A, and B to be CW-complexes with X's basepoint in both A and B. the "triad homotopy groups" of this triple will be defined by the ugly formula pi_n(X, A, B) = pi_{n-1}(P(X, B), P(A, A cap B)) there's an l.e.s associated to these objects whose proof is exactly the same as for relative homotopy: ... -> pi_n (A, A cap B) -> pi_n (X, B) -> pi_n(X, A, B) -> ... if the lower triad homotopy groups vanish up through n, then the map pi_r(A, A cap B) -> pi_r (X, B) will be a monomorphism for r = 1, an isomorphism up to n, and an epimorphism at n finally, getting around to an application of simplicial approximation, if (A, A cap B) is an n-connected relative complex and (X, B) is m-connected, then pi_r(A, A cap B) -> pi_r(X, B) will be an isomorphism for 1 <= r < n + m and epic at n + m and that theorem is basically enough to prove Whitehead's theorem: if f: X -> Y is a weak homotopy equivalence of CW-complexes (i.e. pi_i f is an isomorphism for all i), then X and Y are actually homotopic i'll pause to let that sink in one last SAL application: using SAL and local finiteness, we can show that pi_n (wedge_\alpha X_\alpha) = \coprod_\alpha pi_n X_\alpha where \coprod is the free product for n = 1 and direct sum for n > 1 so, for instance, we finally have a tool to prove that pi_1 (S^1 wedge S^1) is the free group on two generators, something i mentioned ages ago so now that i've bored you all to death with topology, let's get back to algebraic topology an immediate application of SAL is that pi_n X is determined by X's n skeleton (corresponding to generators) and the n+1 cells attached in the construction of X's n+1 skeleton (corresponding to relators) this should smack of the RP^2 example i gave above this means we can construct cell complexes with arbitrary homotopy groups given a group presentation consisting of a bunch of generators G and a bunch of relators R, we can let X's i-skeleton be a single point up to n, then at n attach a bunch of n-spheres (again, corresponding to generators) in the n-skeleton at the n+1 stage, we attach a series of n+1 cells with boundaries lying along homotopy classes corresponding to our relators, effectively making those relator elements null-homotopic from there, we just keep sticking in more higher dim'l cells to kill off the higher sphere homotopy groups that we don't want calling the group by the name H, this is denoted K(H, n) and called "an eilenberg-maclane space" if we do this much more carefully, we can actually construct a CW-complex with homotopy groups matching the homotopy groups of some space X along with a map from our constructed complex to the space that's a weak homotopy equivalence combining this with whitehead, there is a unique homotopy class of CW complexes for any one particular sequence of homotopy groups did anyone solve the fibration problem i posed at the end of the last lecture? i'll take that as a no. well, let's solve it now we had a map f: A -> B and we wanted to come up with a sequence of maps A -> X -> B such that A -> X was a homotopy equivalence, X -> B was a fibration, and the composition was f the solution to this, which isn't hard to see but is difficult to come up with, is to take X to be a subspace of the (non-pointed) pathspace of B cross A such that (gamma, a) satisfies gamma(0) = f(a) the map A -> X is the one that sends a to (the map that sits at f(a), a) for the same reason that the pathspace deformation retracts to a point, X deformation retracts to A -- just suck the paths in finally, the map X -> B is given by evaluating a path at its far endpoint, and that map is a fibration again for the same reason the pathspace evaluation is a fibration we fix a space X and construct a space X whose homotopy groups below n vanish and whose homotopy groups at and above n are the same as X's using the eilenberg-maclane construction outlined above and then we convert the map X -> X given to us by construction into a fibration this particular fibration we call the n-connected cover of X if you go through the l.e.s. associated to this fibration, you'll find that the homotopy groups of X come from two places those above n will come from the connected cover, those below n will come from the fiber, which looks like a coproduct of eilenberg-maclane spaces finally, the sequence ... -> X -> X -> ... -> X<1> -> X is called the "postnikov tower" of X, for those interested enough to look this up elsewhere for those scratching their heads, wondering why i'm bringing this up, recall that the Serre spectral sequence revolves around fibrations, and so introducing as many of them as possible is important (also my research involves H_* BU<2k>) and with that mass of machinery out of the way, it's time to talk about homology, which should be a breath of fresh air any questions before i plow onwards? okay. so now we've seen some more serious things involving homotopy, and it seems like the playing field is just getting more and more cluttered having to keep track of connectedness, and some things are only isomorphisms for a while before they fail miserably, all that is cumbersome in part, this is because topology is still heavily involved in studying these constructs; homotopy is still about continuous maps of things the benefit is that we're still close to the topology we wish to solve and we have some amount of geometric intuition homology comes in at the other end of the spectrum. what if all our theorems were pretty? what if we sacrificed being a first cousin to topology? there are a bunch of different kinds of homology, and i'm going to construct one that fits with the CW-complex theme of today: cellular homology remember that to a CW-complex we associate a bunch of 0 cells, 1 cells, 2 cells, on and on we take a free Z-module on these cells, grading by the dimension of the cell in addition, we attach each n-cell along a (bunch of) (n-1)-cells, and how many times we wander around these cells matters, as exhibited by RP^2 we can formalize exactly how many times we walk around cells as follows: let e^{n-1} be an n-1 cell in the n-1 skeleton of X, and let f: bd. D^n = S^(n-1) -> S^(n-1) be the map given by taking the attaching map of our n-cell and quotienting together the entire (n-1)-skeleton except for the (n-1) cell we've picked out this induces a map pi_{n-1} S^(n-1) -> pi_{n-1} S^(n-1) in homotopy, where pi_{n-1} S^(n-1) = Z and so the map pi_{n-1} f acts like multiplication by some integer. that integer we call the "degree" of f we fit this into our big free module as follows: we construct a map C_n -> C_{n-1} that takes an n-cell in C_n to sum_alpha (deg f_alpha) e_alpha^{n-1} -- i.e. to a formal sum of cells it attaches to, weighted by how many times it wraps around them this map satisfies the property that C_{n+1} -> C_n -> C_{n-1} is the zero map for all choices of n and this bunch of modules and bunch of maps we call a 'chain complex' as you'll recall from my half-proof of the exactness of the l.e.s of relative homotopy, showing that the composition of two maps is zero is only half of the proof that a sequence is exact that demonstrates that im g <= ker f; you must also show that ker f <= im g in general, our chain complex won't satisfy that, but we can measure its failure to do so by replacing C_n with the kernel of the map going out of C_n modulo the image of the map coming in, called the homology at n somewhat incredibly, these homology groups do actually describe something about the space, and satisfy a myriad of properties for instance, this assignment is functorial, and it satisfies the wedge axiom (H_* wedge_alpha X_alpha = sum_alpha H_* X_alpha) we can form relative homology groups for a pair (X, A) by quotienting C_n(X) by C_n(A). in this context, there's a condition on a triad (X, A, B) such that H_*(X, Y) = H_*(X \ U, Y \ U), and there's also a relative l.e.s H_* X -> H_* Y _> H_*(X, Y) proofs of all this is pretty lengthy, and they can all be found in the second chapter of Hatcher's free book _Algebraic Topology_ we can also introduce arbitrary coefficient groups (instead of just Z-modules) by tensoring the sequence ... -> C_{n+1} -> C_n -> C_{n-1} -> ... with our coefficient group G investigating how this affects the homology of a space involves some homological algebra and the Tor functor we can also flip all the arrows and look at something we could study instead of homology: cohomology pick a group G and image the chain complex by Hom_{Ab}(-, G) this induces a codifferential on the resultant sequence of modules that acts by (delta f)(sigma) = f(delta sigma), where the delta on the left is the codifferential and the delta on the right is the old differential again, these satisfy delta^2 = 0, and so we quotient in the same way to measure inexactness so, why not compute some homology groups? a sphere's a good place to start, let's try a 3-sphere since we know that has really ugly homotopy groups as we go higher up a 3-sphere is constructed from a 0-cell and a 3-cell, and so all the maps in our chain are zero and so taking homology groups doesn't change anything, and we're left with a Z in degree 3 and in degree 0 and that's it! all the homology groups of the sphere ah, fuck, i've got to run. as exercises: look up the mayer-vietoris sequence and the hurewicz theorem, the difference between reduced and unreduced theories and how they might fit together (think "relative to a point"), and as an exercise pick your favorite CW complex and compute its homology and cohomology i suggest a torus and a punctured torus my apologies, i'll be back in a few hours, i can answer lingering questions then :( pause? or fin? fin, likely so the excercise is for next week? next time we'll finish introducing homology, ideally by appealing to spectra, then finally construct a spectral sequence and use it a few times. maybe calculate H^* BU<2k> for small k or something yes, the exercises are for next week, mostly with intent to look at things i ran out of time to talk about Category:Seminar